Heavy hitters via cluster-preserving clustering

In turnstile $\ell_p$ $\varepsilon$-heavy hitters, one maintains a
high-dimensional $x\in\mathbb{R}^n$ subject to $\texttt{update}(i,\Delta)$
causing $x_i\leftarrow x_i + \Delta$, where $i\in[n]$, $\Delta\in\mathbb{R}$.
Upon receiving a query, the goal is to report a small list $L\subset[n]$, $|L|
= O(1/\varepsilon^p)$, containing every "heavy hitter" $i\in[n]$ with $|x_i|
\ge \varepsilon \|x_{\overline{1/\varepsilon^p}}\|_p$, where $x_{\overline{k}}$
denotes the vector obtained by zeroing out the largest $k$ entries of $x$ in
magnitude.
For any $p\in(0,2]$ the CountSketch solves $\ell_p$ heavy hitters using
$O(\varepsilon^{-p}\log n)$ words of space with $O(\log n)$ update time,
$O(n\log n)$ query time to output $L$, and whose output after any query is
correct with high probability (whp) $1 - 1/poly(n)$. Unfortunately the query
time is very slow. To remedy this, the work [CM05] proposed for $p=1$ in the
strict turnstile model, a whp correct algorithm achieving suboptimal space
$O(\varepsilon^{-1}\log^2 n)$, worse update time $O(\log^2 n)$, but much better
query time $O(\varepsilon^{-1}poly(\log n))$.
We show this tradeoff between space and update time versus query time is
unnecessary. We provide a new algorithm, ExpanderSketch, which in the most
general turnstile model achieves optimal $O(\varepsilon^{-p}\log n)$ space,
$O(\log n)$ update time, and fast $O(\varepsilon^{-p}poly(\log n))$ query time,
and whp correctness. Our main innovation is an efficient reduction from the
heavy hitters to a clustering problem in which each heavy hitter is encoded as
some form of noisy spectral cluster in a much bigger graph, and the goal is to
identify every cluster. Since every heavy hitter must be found, correctness
requires that every cluster be found. We then develop a "cluster-preserving
clustering" algorithm, partitioning the graph into clusters without destroying
any original cluster.